The space \(M(\Sigma )\,=\, G^{2(r+g)}\)  with coordinates \( (a_i,\, b_i,\, u_j ,\, v_j)\) 
for \(i\, =\,1,\, \dots ,\, g\) , and \(j \,= 1,\, \dots ,\, r\) .
The action of \((z_0,\, \dots ,\, z_r)\,\in \, G^{r+1} \)  is given by (see [1]}, Equation (38))
\(  a_i\,\longmapsto \, {\rm Ad}_{z_0} a_i, \) 
\( b_i\,\longmapsto \, {\rm Ad}_{z_0} b_i ,\) 
\(u_j \,\longmapsto \, z_0 u_j z_j^{-1} \) 
\(v_j \,\longmapsto \, {\rm Ad}_{z_j} v_j\) 
